Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain

We present a computational method for reconstructing a vector field on a convex polytope $\mathcal{P}\subset {\mathbb{R}}^d$ of arbitrary dimension from discrete samples. We specifically address the scenario where the vector field is subject to a no-penetration (slip) boundary condition, requiring it to be tangent to the boundary $\mathrm{\partial }\mathcal{P}$. Given a degree bound $k$, our algorithm computes a polynomial vector field of degree at most $k$ that fits the observed data in the least-squares sense while exactly satisfying the tangency constraints. Central to our approach is an explicit characterization of the module of polynomial vector fields tangent to $\mathrm{\partial }\mathcal{P}$, derived using algebraic concepts from the theory of hyperplane arrangements.